Above a certain level, not very important. IQ is a hygiene factor, not a discriminating factor: it helps to be reasonably smart, but above a certain threshold, further increments help less than in some other fields. The two PhD’s in the book specifically commented on this…

Sushil, Chapter 5: “…for anyone in the top few percent of the population, IQ points are not the limiting factor in investment – other qualities such as recognising your ignorance matter more.”

Vernon, Chapter 7: ” Compared to his mathematics degree and his doctoral work in computer science , he thinks investing is not technically difficult, but it does require some mental habits which are possibly unusual, although not difficult…”

Emanuel Derman gives another version of this: "I once said to one of our equity traders that I thought, on average, fixed-income traders seem to be smarter than equity traders. He agreed, adding ‘That’s because there’s no competitive advantage to being that smart in equity trading’."

I think it helps to distinguish between mental strength and mental characteristics.

Indifference to group affiliation. (Buffett in 1965: “We derive no comfort because important people, vocal people, or great numbers of people agree with us. Nor do we derive comfort if they don't.” )

For intelligence above a certain level, mental characteristics are related only weakly, if at all, to IQ.

By the way, the mental characteristics listed above are not universally positive. There are many occupations and endeavours which require the protagonists to maintain habitual biases (eg politics), reduce or deny uncertainty (eg leadership), and affirm group affiliation (eg almost any role in any organisation). The mental characteristics which are helpful for investing may not be much use for anything else.

"He drew a distinction between activities with ‘positive scoring’, where success is defined by gaining wins, and activities with ’negative scoring’, where success is defined by avoiding faults.

Positively scored activities include selling, leadership, and most sports. In these activities bravery, ‘having a go’ and risk-taking give a better chance of success than careful deliberation, and the downside of making errors is low. Negatively scored activities include driving a car, piloting an aircraft, and anaesthetics in medicine. The successful driver, pilot or anaesthetist is not the brave one who always ‘has a go’, but rather one who never makes any big mistakes."

Here's a more succinct, more mathematical way of putting this. For any activity, what's your payoff function - the max, the min or the average?

Max payoffs(Vernon's "positively scored" activities) Your payoff is your highest result. Failure has no lasting consequences. High energy, irrational optimism and persistence are optimal traits. Don't worry about failure, just get on and "have a go."

Investment is an activity with weighted average payoffs. But it helps - both because of the maths of compound growth (see earlier posts) and for psychological reasons - to place particular weight on the min payoffs. A lot of success in investing comes from just avoiding mistakes.

Another metaphor for this idea is winner's games and losers' games. Tennis played between expert professionals is a winner's game: the winner is the player who hits the most winning shots. Tennis played between mediocre amateurs is a loser's game: the winner is the player who makes the fewest unforced errors. In this sense, investment is probably a loser's game.

Chapter 5, Sushil, talks about optimal leverage and compound growth. The main takeaway is that investors are better off maximising expected logarithmic return (not expected return), and this often means lower leverage than you think.

In my view expected return (the mean) is often a poor objective, because the distribution of terminal wealth from any long period of compounding is likely to be very positively skew – a small number of very high outcomes contribute a lot to the mean. Median terminal wealth – found by summing the logarithmic returns, and then taking the anti-logarithm – seems a better measure of ‘typical’ results than the mean.

For example, consider an investment which either doubles or falls 60%, each with probability 50% in each period, over a time horizon of 3 periods. This gives eight equi-probable outcomes for terminal wealth levels: 0.064, 0.32 (three times), 1.6 (three times), 8. Taking the mean of these, we find mean terminal wealth as 1.728 (=1.23). If you used the mean as your objective, the investment looks good. But 7 of 8 possible outcomes are below this. The median terminal wealth of exp –{0.11x3} = 0.72 seems a better measure of 'typical' results.

If we extend the scale of the graph in the book, we can show the expected compound growth (the blue line) as well as median compound growth (the red line). Adding leverage always increases the expected (the blue line slopes upwards), but increases the median only up to a certain point L* (the red line has a maximum). And with too much leverage, the red line eventually goes negative - which means that in the long term, you almost surely go broke.

We can also draw a graph for the example Sushil talks about: an investment which rises 25% or falls 20% in each period. With no leverage (L = 1), the expected compound growth is +2.5% per period (blue line intercept on the y-axis), but median compound growth is zero (red line). The optimal leverage in this case is L* = 0.5, that is, invest only half your bankroll, keeping the other half in cash (assumed nil return here, for simplicity).

Of course, these examples – repeated trials of investments with only two equi-probable outcomes in each period - don’t correspond to any real-world problem. The value of the examples is to highlight the following points.

mean return is what casual intuition leads to, but this is not the same as median return, which is usually lower

median return is the best measure of 'typical' results over long periods

if we want to maximise median return, we need to maximise expected logarithmic return.

Looking up logarithms is inconvenient for mental arithmetic, so it helps to have a simpler approximation. Suppose E is the expected return of the investment, and V is the variance of return. For short periods E will typically be small, so LE will be small, and log(1+LE) ≈ LE, and (1+ LE)2 ≈1; and if we use these approximations for the first and second terms of the Taylor expansion of log (1+LE), the errors in the two approximations are of opposite sign. A quick mental arithmetic approximation of the expected log return of a leveraged investor is then

LE – L2 V / 2 .

This expression is maximised when L = E / V. For example, suppose you invest in equities with an expected return (net of any charges) of 6%pa and standard deviation of return of 20%pa (ie V = 0.22 = 0.04). The theoretically optimal leverage is 0.06/0.04 = 1.5x. In practice I would halve the leverage above 100% for safety, ie maximum 1.25x, (partly because I don't know if my estimates of E and V are correct). Leverage of L > 3x gives negative expected log return, ie you would almost surely go broke in the long run. Quite a few hedge funds operate with leverage > 3x !

This quick estimate (LE – L2 V / 2) for the expected log return is often enough to highlight the lack of safety in common financial structures, including some cases of hedge funds, spread betting and CFD leverage, and split capital investment trusts.

Over the years I have observed many successful private and professional investors. There have been a very few – including some, but not all, of the investors in Free Capital – about whom I’ve thought “If I ever get bored of investing, I'd be happy for them to manage my money.”

Most of these people have eventually disappointed me. The disappointments have tended to arise because they got stuck with concentrated positions – 10% of one small company, 20% of another – and were unable to react when the world changed against the businesses to which they had made large commitments.

This affects even the very greatest investors. Look at Warren Buffett: large positions in the Washington Post and other print newspapers, businesses which now look a lot like manufacturers of horse-drawn carriages, circa 1900.

Having seen many people whom I greatly admired come a cropper with concentrated positions, I’ve become less enamoured of extreme concentration. Nothing grows forever; most investors come a cropper eventually; and concentrated positions in illiquid shares make it hard to escape from your mistakes.

(2) Some practicalities

My last post discussed how many shares one should hold in a relatively abstract way. Another angle is to ask how many shares it is practical to follow.

There is clearly some sort of trade-off between number of shares held and your quality of knowledge. Many experienced private investors are surprised (and often derisory) about the fact that that I hold around 50 shares. Surely I must know far less about many of them than if I held only 10 or 15 shares?

Well, perhaps. But this is diamonds versus flower bulbs again. If I held 10 or 15 shares, I would want them all to be diamonds. I find diamonds very hard to recognise, and most of the time, I can't find 10 or 15 of them. A lot of the time, I can't find any of them.

Flower bulbs much easier to find. At most times I can find some shares on P/E ratios around 5, shares trading below net cash, shares where there’s been a failed bid but the bidder looks likely to return, or shares with some other idiosyncratic cheapness. The quality of these businesses is often low (they aren’t diamonds). But there are plentiful, and easy to recognise.

My portfolio of 50 flower bulbs comprises businesses of lower average quality than if I held 10 diamonds. But the certainty of my insight about the 50 flower bulbs is relatively high. This is because my insights about flower bulbs tend to be (a) simple and (b) short-term. For example: "it's trading 25% below net cash, and something is likely to happen to make it trade at net cash."

Insights about diamonds tend to be more complex, long-term and nebulous. "It has long-term comparative advantage and it's going to be a bigger business in 5, 10 and 20 years' time." So you say. But how do you know? Generally, I don't.

Or to put it another way: to identify diamonds, you need long-term foresight about their durability - market position, susceptibility to technological changes, etc etc. I've always found this very difficult. But you don’t need long-term foresight to identify flower bulbs.

A final point in favour of diversification is that even if a very few shares is the optimal policy in theory, many highly concentrated investors eventually come a cropper in practice.

(1) Some theory

Most of the investors in Free Capital hold concentrated portfolios, sometimes of fewer than ten shares (Luke, Owen and Taylor). Others such as Eric, John Lee, Peter Gyllenhammar and Sushil hold up to 60 shares. Who is right? How many is too many?

Many experienced investors advocate a small number of holdings. I used to think this was always good idea. I still think this is probably right for a knowledgeable investor with a relatively small fund. But for an investor managing millions or tens of millions of pounds (like most people in the book), I'm not so sure.

If N is my number of stocks to hold, I think the optimal N is a function of the following variables...

N = f { quality of knowledge about return dispersions (↓),

£ size of portfolio (↑)

volatility of shares (↑)

capital gains tax rate (↓) }

...that is, a decreasing function of quality of knowledge and CGT rate, and an increasing function of the £ size of your portfolio and the volatllity of the candidate shares.

Let me explain....

Decreasing function of quality of knowledge about return dispersions This point is fairly obvious. If you know with certainty which share in the market will give the best return over your time horizon, all your portfolio should be in that one share(*). If you know nothing about the return dispersions the optimal portfolio is indexation. In reality, most of use are somewhere in between. Exceptional investors with exceptional quality of knowledge should hold a relatively concentrated portfolio.

For example, most years between 1977 and 2000, Warren Buffett appears to have held around one-third of his portfolio in his largest holding (usually a different holding in each successive year). I don’t hold one-third of my portfolio in one share, because I’m not that good.

(* Side-note This “obvious” point may not be mathematically correct. See information theory: if transaction costs are zero, a constantly rebalanced universal portfolio must asymptotically out-perform the best-performing share in the market. Of course, in practice transaction costs are never zero. But the point illustrates that the question discussed in this post is quite subtle; observations which are "obvious" are not necessarily correct.)

Increasing function of £ size of portfolio With a small portfolio, liquidity is not a constraint. If you know with certainty which share will give the best return over a time horizon, you can put all your money in that one share and sell at the end of the time horizon. With a larger portfolio, liquidity becomes a constraint. Even with perfect knowledge, you can no longer "buy at the bottom and sell at the top."

With a larger portfolio, a larger number of holdings becomes optimal – not because it “spreads risk”, but because it increases liquidity, and so increases options to change your mind as prices and your expectations change.

For example, if I invest £50,000 in each of my 10 best smallcap ideas, I can sell any which rise to be fully valued. If I invest £500,000 in my single best smallcap idea, and it rises to be fully valued, I probably won’t be able to sell all of it anywhere near that price (who is going to buy?). Even with perfect knowledge dispersion of returns, it’s attractive to invest some in the 2nd, 3rd, 4th… best ideas, and thus create options to sell shares when they go up, and buy when they go down.

Increasing function of the volatility of candidate shares If the candidate shares are highly volatile, it becomes attractive to switch frequently, buying individual shares when they are low and selling when they are high (volatilty pumping). To do this, you need liquidity, that is you need to restrict your size in any one share, to ensure you can“buy at the bottom and sell at the top." For given £ size of portfolio, smaller size in any one share implies a larger number of shares.

Decreasing function of the CGT rate If the CGT rate is high, the tax penalty on turnover is high, so the optimal portfolio probably has low turnover. You should pick only shares which you are happy to hold for 10 or 20 years (I call these “diamonds” – see last post); this probably means very few shares indeed, and these shares are difficult to identify (ie the chance of "false positive" errors is high).

On the other hand if the tax rate is zero, there is no need to attempt the difficult and error-prone search for a very few 20-year diamonds. You can instead buy “flower bulbs” (see last post), which are always more plentiful than diamonds, and much easier to recognise. Flower bulbs can only be bought in small size, because you need to be able to sell easily and promptly when the flower blooms. For a given £ size of portfolio, smaller size for individual shares implies a larger number of holdings.

Diamonds These are the shares orthodoxy says you should buy: shares in businesses with exceptional economics and long-term comparative advantage. They are shares you would buy if you followed the Buffett doctrine: choose shares you would be happy to hold if the stock market closed tomorrow for five years.

Flower bulbs These are shares which are cheap just at the moment, but without any exceptional long-term quality. Flower bulbs trade on a P/E ratio of 5, or have net cash per share in excess of the share price, or some other idiosyncratic cheapness – together with no negative “hygiene factors” (see Vernon, Chapter 7). Flower bulbs can usually be relied upon to bloom, but they don’t have any exceptional long-term qualities. A flower bulb can be a good buy, but only in modest size: you need liquidity to sell when the flower blooms.

Which should you buy, diamonds or flower bulbs? I used to think diamonds, but I’ve come to realise that real diamonds are rare, and very hard to distinguish from fakes. To recognise a diamond, you need long-term foresight of its durability. I’ve always found this very difficult. My error rate for false positives as a diamond assessor is too high.

Flower bulbs are much more common than diamonds, and easier to recognise. You don’t need long-term foresight. You just need to recognise something that is going to bloom, and keep your holding small enough to sell when it does.

I’ve largely given up looking for diamonds. Nowadays, I spend much of my time scavenging the dustbins of the stock market for flower bulbs someone has thrown away by mistake. Less glamorous than prospecting for diamonds, but more reliable. Remember Bill, Chapter 3: "Investing is a field where knowing your limitations is more important than stretching to surpass them.”

Operating his own bulletin board, Nigel has thought about the conceptual ingredients of successful online discussion communities. Most popular boards have most of the following properties.

Pseudonymity:Many people do not want their postings, particularly on a sensitive topic such as personal investments, to be searchable by employers or casual acquaintances, or even family members.

Persistent identity: Although pseudonymity is desirable for privacy reasons, any meaningful discussion amongst users requires some form of persistent identity, that is, a stable one-to-one mapping of aliases to users.